1 Answer
1

A lot.

We don't get exact numbers, but the chapter The Letters From No One gives us some idea of the number of letters involved. It's in the order of a few hundred:

When Harry receives his first letter:

Yet here it was, a letter, addressed so plainly there could be no mistake.

The following day, there's a rush for the post, and he receives a second letter:

When the post arrived, Uncle Vernon, who seemed to be trying to be nice to Harry, made Dudley go and get it. They heard him banging things with his Smeltings stick all the way down the hall. Then he shouted, “There’s another one! Mr H. Potter, The Smallest Bedroom, 4 Privet Drive –”

The following day, Harry tries to wake up early to get the post. He finds that Vernon had the same idea, and slept on the doormat:

Harry shuffled miserably off into the kitchen, and by the time he got back, the post had arrived, right into Uncle Vernon’s lap. Harry could see three letters addressed in green ink.

By Saturday, somebody at Hogwarts has realised the letters aren't getting through, and decides to get creative:

On Saturday, things began to get out of hand. Twenty-four letters to Harry found their way into the house, rolled up and hidden inside each of the two dozen eggs that their very confused milk-man had handed Aunt Petunia through the living-room window.

On the Sunday, the kitchen is flooded with letters – this scene is reproduced in the film. Somebody really wants to write to harry:

Something came whizzing down the kitchen chimney as he spoke and caught him sharply on the back of the head. Next moment, thirty or forty letters came pelting out of the fireplace like bullets. The Dursleys ducked, but Harry leapt into the air trying to catch one –

“Out! OUT!”

Uncle Vernon seized Harry around the waist and threw him into the hall. When Aunt Petunia and Dudley had run out with their arms over their faces, Uncle Vernon slammed the door shut. They could hear the letters still streaming into the room, bouncing off the walls and floor.

Vernon takes them to a hotel, thinking they'll leave the letters behind them at Privet Drive. Hogwarts is not so easily deterred:

They ate stale cornflakes and cold tinned tomatoes on toast for breakfast next day. They had just finished when the owner of the hotel came over to their table.

“ ’Scuse me, but is one of you Mr H. Potter? Only I got about an ’undred of these at the front desk.”

Finally, he takes them beyond the reach of the conventional Muggle postal system, to a hut on a barren island. But Hagrid is not a conventional postman. He delivers Harry's final letter:

Harry stretched out his hand at last to take the yellowish
envelope, addressed in emerald green to Mr H. Potter, The Floor,
Hut-on-the-Rock, The Sea.

We know he got at least about 106. The question is how many came out of the kitchen chimney. We know roughly 30-40 came out on the first second, so at least 140 overall. If we then assume those first 30-40 were bottled up and only about 5 per second after that then we only need to know how long it lasted to get a rough total.
– XantecOct 5 '15 at 14:12

8

"their very confused milk-man had handed Aunt Petunia through the living-room window" The milk-man is confused because he is delivering the eggs by the quite unconventional method of handing them through the living-room window, rather than through a more traditional door (which were all boarded up to prevent the letters from coming in, or so the Durselys thought...)
– Michael BeckerOct 5 '15 at 17:24